354: Area of Hyperrectangle Can Be Approximated by Area of Covering Finite Number Hypersquares to Any Precision

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A description/proof of that area of hyperrectangle can be approximated by area of covering finite number hypersquares to any precision

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Topics

About: Euclidean metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

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Target Context

• The reader will have a description and a proof of the proposition that the area of any hyperrectangle can be approximated by the area of covering finite number hypersquares to any precision.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

The area, $a$, of any hyperrectangle on any ${\mathbb{R}}^n$ Euclidean metric space can be approximated by the area of some finite number of hypersquares, $a_i$, that cover the hyperrectangle, to any precision, which is, for any real number $ϵ>0$, $\sum _i{a}_i-a<ϵ$.

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2: Proof

Denote the side lengths of the hyperrectangle as $l_1,l_2,...,l_n$. For any real number, $l_s>0$, and each side index, $1\le i\le n$, there is the non-negative unique integer, $k_i(l_s)$, and the unique real number, $0\le l_i^{\prime }(l_s)<l_s$, such that $k_i(l_s)l_s=l_i^{\prime }(l_s)+l_i$, which means that the $l_i$ length is covered by $k_i(l_s)$ times $l_s$ with the excess, $l_i^{\prime }(l_s)$, where the unique existences are clear based on the meaning.

Now, $l_s$ is the side length of the same-size hypersquares that cover the hyperrectangle, starting from a vertex of the hyperrectangle, $k_i(l_s)$ times for each $i$ direction, with the excess of $l_i^{\prime }(l_s)$ for the $i$ direction.

The excess area is the error of the approximation, which is $e(l_s)=\prod _i{l}_s{k}_i(l_s)-\prod _i{l}_i$. But by directly calculating the excess area, $e(l_s)<\sum _i{l}_i^{\prime }(l_s)\prod _{j\ne i}(l_j^{\prime }(l_s)+l_j)$, adding some areas multiple times. Assuming $l_s\le 1$ (which we are free to do) , $\sum _i{l}_i^{\prime }(l_s)\prod _{j\ne i}(l_j^{\prime }(l_s)+l_j)<\sum _i{l}_s\prod _{j\ne i}(l_s+l_j)<l_s\sum _i\prod _{j\ne i}(1+l_j)$.

As $L:=\sum _i\prod _{j\ne i}(1+l_j)$ depends only the configuration of the hyperrectangle, we choose $l_s$ as $min(1,\frac{ϵ}{L})$, then, $e(l_s)<l_{s}L\le ϵ$.

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References

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